In a prior art embodiment of the gas connection device of a conventional insufflation device, the outlet port is mounted in the device wall, and is connected, over a copper pipe, with the pressure reducer. The latter is connected, over a pressurised-air hose, with the safety valve, and over another copper pipe, with a pressure-indicating manometer. Finally, another pressurised-air hose is present between the safety valve and the solenoid valve. The copper pipes and the pressurised-air hoses are connected over screwed joints with special cap nuts, to the respective components of the gas connection device. Thereby, when mounting the prior art gas connection device within the housing of an insufflation device, several individual assemblies have to be installed, and have to be connected with each other over copper pipes and pressurised-air hoses. Processing of the copper pipes requires an enormous expenditure by cutting, bending, chromating, removal of chips and the like. Further, considerable costs are caused by stocking and purchasing the copper pipes, pressurised-air hoses, screwed joints and cap nuts. In total, there are relatively high assembly costs and a large number of potential failure locations.